https://en.wikipedia.org/wiki/Reeb_graph

https://en.wikipedia.org/wiki/Reeb_graph http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

mapper.filters.kNN_distance(data, k, metricpar={}, callback=None) The distance to the k-th nearest neighbor as an (inverse) measure of density. Note how the number of nearest neighbors is understood: k=1, the first neighbor, makes no sense for a filter function since the first nearest neighbor of a data point is always the point itself, and hence this filter function is constantly zero. The parameter k=2 measures the distance from xi to the nearest data point other than xi itself.

mapper.filters.kNN_distance(data, k, metricpar={}, callback=None) The distance to the k-th nearest neighbor as an (inverse) measure of density. Note how the number of nearest neighbors is understood: k=1, the first neighbor, makes no sense for a filter function since the first nearest neighbor of a data point is always the point itself, and hence this filter function is constantly zero. The parameter k=2 measures the distance from xi to the nearest data point other than xi itself. d1(x) = d2(x) = d3(x) = d4(x) = z 5 4 x y 3

x y If x is in a denser region than y, then dk(x) dk(y)

Image of a circle under linear transformation T(x) = Ax where A is a symmetric matrix

https://en.wikipedia.org/wiki/File:Singular-Value-Decomposition.svg The SVD decomposes M into three simple transformations: a rotation V*, a scaling Σ along the coordinate axes and a second rotation U.

mapper.filters.dm_eigenvector(data, k=0, mean_center=True, metricpar={}, verbose=True, callback=None) Return the k-th eigenvector of the distance matrix. The matrix of pairwise distances is symmetric, so it has an orthonormal basis of eigenvectors. The parameter k can be either an integer or an array of integers (for multi-dimensional filter functions). The index is zero-based, and eigenvalues are sorted by absolute value, so k=0 returns the eigenvector corresponding to the largest eigenvalue in magnitude. If mean_center is True, the distance matrix is double-mean-centered before the eigenvalue decomposition. Reference: [R6], subsection “Principal metric SVD filters”.